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It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position.

Is there a known method to produce a reasonable lower bound on the bridge index?

For example, knot 11a1 has at most bridge index $4$, because a $4$-bridge position of this knot can be drawn: DT code (42, 66, 44, 56, 54, 46, 64, 40, -38, 86, 94, 76, 100, 80, 82, 98, 74, 96, 84, -110, -108, -68, -104, -22, -34, -32, -24, -102, -26, -30, -36, -20, -106, 90, 122, 60, 50, -116, -10, -6, -120, -70, -112, -14, -2, -4, -12, -114, -72, -118, -8, 78, 92, 88, 16, 18, 62, 48, 52, 58, -28) gives one such diagram. What is known about methods to eliminate the possibility that this knot has a $2$-bridge or $3$-bridge projection?

Update

After posting this question I found a $3$-bridge projection (and because the knot is not rational, this is the lowest possible projection), given by DT code (12, 16, 58, 60, 14, -92, -90, -94, -32, -40, 120, 102, 108, 112, 98,116, 124, 106, 104, 122, 118, 100, 110, -26, -34, -38, -22, -42,-30, -96, -28, -44, -24, -36, 50, 18, 56, 62, 46, 64, 54, 20, 52, 66, 48, 128, 126, 114, -6, -74, -80, -86, -68, -88, -78, -76, -8, -4, -72, -82, -84, -70, -2, -10), but I am still interested in what is already known about establishing lower bounds more generally. Ryan Budney's suggestion of the Heegaard genus is a good one, but I haven't found a reference that shows this bound to be sharp.

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 There are lower estimates for the bridge number in terms of the Jones polynomial, or more generally in terms of TQFT invariants. In principle, there should be an algorithm to compute the bridge number analogous to an algorithm to compute Heegaard genus (using almost normal surfaces), but I'm not sure anyone has written it up. – Agol Dec 19 2011 at 19:12

2 Answers

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I imagine bridge index is readily computable, at least for "small enough" knots. Bridge index is a lot like Heegaard genus of a 3-manifold. Heegaard splitting surfaces can be found via (almost) normal surface theory, provided you have a triangulation of the 3-manifold, and the triangulation isn't so big that the search for the surface doesn't take too much memory or time.

Bridge index is analogous to Heegaard genus for closed manifolds. In that, if a knot has bridge index $k$ there is a $2k$-punctured sphere properly embedded in the complement, separating the complement into the union of two 3-manifolds that are handlebodies. Heegaard surfaces can generally be found via almost normal surface theory, so presumably the bridge index can be found in an analogous way. Off the top of my head I don't know the proper references for this but I believe the results for Heegaard splittings go back to Rubinstein (and perhaps other results using normal surface theory that go back to Waldhausen). If you don't get an answer here in the next few days I can ask someone who likely knows.

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 You get a lower bound by considering the Heegaard genus of the branched double cover: is it always tight? (I don't see any obvious reason why it should be, though) – Marco Golla Dec 8 2011 at 8:53
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 The equivariant Heegaard genus would be a tight relation, but presumably that's not the same thing as the Heegaard genus. The problem with this avenue is it says one hard-to-compute thing is related to another hard-to-compute thing. IMO if you really want to compute the bridge index, rather than just relate it to a lot of other hard to compute things your best bet with today's techniques would be normal surface theory. – Ryan Budney Dec 19 2011 at 7:25
Thanks, I appreciate the thought. – Chad Musick Dec 19 2011 at 8:49
Ryan: I completely agree with everything you said, with the proviso that normal surface algorithms become very tricky in the presence of normal tori. (The basic issue is: you want to enumerate things of low complexity, where negative Euler characteristic is your basic measurement of complexity. When you have a normal torus, other surfaces can "spin" around it, meaning the number of low-complexity surfaces in your manifold is unbounded.) So normal surface algorithms will work well if the knot complement is hyperbolic (this is the case in Coward's preprint) and not as well otherwise. – Dave Futer Dec 19 2011 at 21:04
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This is really a comment on Ryan's answer, but I don't have enough rep. Here are two relevant papers: http://front.math.ucdavis.edu/0710.1262 (Algorithmically Detecting the bridge number for hyperbolic knots by Alex Coward)

and

http://front.math.ucdavis.edu/0709.3534 (Meridional Almost Normal Surfaces in Knot Complements by Robin Wilson)

I know of some people who are working on other ways to get lower bounds for bridge number in terms of other invariants.

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Thanks. I had read Coward's paper before, but I hadn't seen Wilson's. – Chad Musick Dec 19 2011 at 21:05
I'm pretty sure Alex Coward's preprint is still the state of the art today. – Dave Futer Dec 19 2011 at 21:05
Scott, hopefully with all these upvotes you'll have the ability to add comments soon! Thanks for filling in the gaps in my own answer. – Ryan Budney Dec 20 2011 at 0:01

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